Granular Matter

, Volume 16, Issue 4, pp 509–515 | Cite as

Shear dispersion in dense granular flows

  • Ivan C. Christov
  • Howard A. Stone
Original Paper


We formulate and solve a model problem of dispersion of dense granular materials in rapid shear flow down an incline. The effective dispersivity of the depth-averaged concentration of the dispersing powder is shown to vary as the Péclet number squared, as in classical Taylor–Aris dispersion of molecular solutes. An extension to generic shear profiles is presented, and possible applications to industrial and geological granular flows are noted.


Taylor–Aris dispersion Rapid granular flow Bagnold profile Granular diffusion 



I.C.C. was supported by the National Science Foundation (NSF) under Grant No. DMS-1104047 (at Princeton University) and by the LANL/LDRD Program through a Feynman Distinguished Fellowship (at Los Alamos National Laboratory). LANL is operated by Los Alamos National Security, L.L.C. for the National Nuclear Security Administration of the U.S. Department of Energy under Contract No. DE-AC52-06NA25396. H.A.S. thanks the NSF for support via Grant No. CBET-1234500. We acknowledge useful discussions with Ian Griffiths and Gregory Rubinstein on the derivation of the dispersion equations for the case of non-constant diffusivity, and we thank Ben Glasser for helpful conversations.


  1. 1.
    Brenner, H., Edwards, D.A.: Macrotransport Processes. Butter- worth-Heinemann, Boston (1993)Google Scholar
  2. 2.
    Taylor, G.: Dispersion of soluble matter in solvent flowing slowly through a tube. Proc. R. Soc. Lond. A 219, 186–203 (1953)ADSCrossRefGoogle Scholar
  3. 3.
    Aris, R.: On the dispersion of a solute in a fluid flowing through a tube. Proc. R. Soc. Lond. A 235, 67–77 (1956)ADSCrossRefGoogle Scholar
  4. 4.
    Young, W.R., Jones, S.: Shear dispersion. Phys. Fluids A 3, 1087–1101 (1991)ADSCrossRefzbMATHGoogle Scholar
  5. 5.
    Natarajan, V.V.R., Hunt, M.L., Taylor, E.D.: Local measurements of velocity fluctuations and diffusion coefficients for a granular material flow. J. Fluid Mech. 304, 1–25 (1995)ADSCrossRefGoogle Scholar
  6. 6.
    Campbell, C.S.: Self-diffusion in granular shear flows. J. Fluid Mech. 348, 85–101 (1997)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  7. 7.
    Savage, S.B.: The mechanics of rapid granular flows. Adv. Appl. Mech. 24, 289–366 (1984)CrossRefzbMATHGoogle Scholar
  8. 8.
    Jaeger, H.M., Nagel, S.R., Behringer, R.P.: Granular solids, liquids, and gases. Rev. Mod. Phys. 68, 1259–1273 (1996)ADSCrossRefGoogle Scholar
  9. 9.
    Andreotti, B., Forterre, Y., Pouliquen, O.: Granular Media: Between Fluid and Solid. Cambridge University Press, Cambridge (2013)CrossRefGoogle Scholar
  10. 10.
    Aranson, I.S., Tsimring, L.S.: Granular Patterns. Oxford University Press, New York (2009)Google Scholar
  11. 11.
    Hacina, A., Kamel, D.: Indirect method of measuring dispersion coefficients for granular flow in a column of dihedrons. Int. J. Food Eng. 4, 10 (2008)CrossRefGoogle Scholar
  12. 12.
    Simsek, E., Wirtz, S., Scherer, V., Kruggel-Emden, H., Grochowski, R., Walzel, P.: An experimental and numerical study of transversal dispersion of granular material on a vibrating conveyor. Particle Sci. Tech. 26, 177–196 (2008)CrossRefGoogle Scholar
  13. 13.
    Iverson, R.M.: The physics of debris flows. Rev. Geophys. 35, 245–296 (1997)ADSCrossRefGoogle Scholar
  14. 14.
    Pudasaini, S.P., Hutter, K.: Avalanche Dynamics. Springer, Berlin (2007)Google Scholar
  15. 15.
    Gray, J.M.N.T., Kokelaar, B.P.: Large particle segregation, transport and accumulation in granular free-surface flows. J. Fluid Mech. 652, 105–137 (2010)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  16. 16.
    Pouliquen, O., Delour, J., Savage, S.B.: Fingering in granular flows. Nature 386, 816–817 (1997)ADSCrossRefGoogle Scholar
  17. 17.
    Nakashizuka, T., Iida, S., Suzuki, W., Tanimoto, T.: Seed dispersal and vegetation development on a debris avalanche on the Ontake volcano. Central Jpn. J. Veget. Sci. 4, 537–542 (1993)CrossRefGoogle Scholar
  18. 18.
    Hwang, C.L., Hogg, R.: Diffusive mixing in flowing powders. Powder Technol. 26, 93–101 (1980)CrossRefGoogle Scholar
  19. 19.
    Savage, S.B., Lun, C.K.K.: Particle size segregation in inclined chute flow of dry cohesionless granular solids. J. Fluid Mech. 189, 311–335 (1988)ADSCrossRefGoogle Scholar
  20. 20.
    Griffiths, I.M., Stone, H.A.: Axial dispersion via shear-enhanced diffusion in colloidal suspensions. EPL 97, 58005 (2012)ADSCrossRefGoogle Scholar
  21. 21.
    Jop, P., Forterre, Y., Pouliquen, O.: A constitutive law for dense granular flows. Nature 441, 727–730 (2006)ADSCrossRefGoogle Scholar
  22. 22.
    Khakhar, D.V.: Rheology and mixing of granular materials. Macromol. Mater. Eng. 296, 278–289 (2011)CrossRefGoogle Scholar
  23. 23.
    Bolster, D., Dentz, M., Le Borgne, T.: Solute dispersion in channels with periodically varying apertures. Phys. Fluids 21, 056601 (2009)ADSCrossRefGoogle Scholar
  24. 24.
    Bagnold, R.A.: Experiments on a gravity-free dispersion of large solid spheres in a Newtonian fluid under shear. Proc. R. Soc. Lond. A 225, 49–63 (1954)ADSCrossRefGoogle Scholar
  25. 25.
    Silbert, L.E., Ertaş, D., Grest, G.S., Halsey, T.C., Levine, D., Plimpton, S.J.: Granular flow down an inclined plane: Bagnold scaling and rheology. Phys. Rev. E 64, 051302 (2001)ADSCrossRefGoogle Scholar
  26. 26.
    Eckstein, E.C., Bailey, D.G., Shapiro, A.H.: Self-diffusion of particles in shear flow of a suspension. J. Fluid Mech. 79, 191–208 (1977)ADSCrossRefGoogle Scholar
  27. 27.
    Leighton, D., Acrivos, A.: The shear-induced migration of particles in concentrated suspensions. J. Fluid Mech. 181, 415–439 (1987)ADSCrossRefGoogle Scholar
  28. 28.
    Vollebregt, H.M., van der Sman, R.G.M., Boom, R.M.: Suspension flow modelling in particle migration and microfiltration. Soft Matter 6, 6052–6064 (2010)ADSCrossRefGoogle Scholar
  29. 29.
    Scott, A.M., Bridgwater, J.: Self-diffusion of spherical particles in a simple shear apparatus. Powder Technol. 14, 177–183 (1976)CrossRefGoogle Scholar
  30. 30.
    Savage, S.B.: Disorder, diffusion, and structure formation in granular flow. In: Hansen, A., Bideau, D. (eds.) Disorder and Granular Media, pp. 255–285. Elsevier, Amsterdam (1993)Google Scholar
  31. 31.
    Christov, I.C., Stone, H.A.: Resolving a paradox of anomalous scalings in the diffusion of granular materials. Proc. Natl Acad. Sci. USA 109, 16012–16017 (2012)ADSCrossRefGoogle Scholar
  32. 32.
    Goldhirsch, I.: Rapid granular flows. Annu. Rev. Fluid Mech. 35, 267–293 (2003)ADSCrossRefMathSciNetGoogle Scholar
  33. 33.
    Savage, S.B., Dai, R.: Studies of granular shear flows: wall slip velocities, layeringx and self-diffusion. Mech. Mat. 16, 225–238 (1993)CrossRefGoogle Scholar
  34. 34.
    Wiederseiner, S., Andreini, N., Épely-Chauvin, G., Moser, G., Monnereau, M., Gray, J.M.N.T., Ancey, C.: Experimental investigation into segregating granular flows down chutes. Phys. Fluids 23, 013301 (2011)Google Scholar
  35. 35.
    Fan, Y., Schlick, C.P., Umbanhowar, P.B., Ottino, J.M., Lueptow, R.M.: Modeling size segregation of granular materials: the roles of segregation, advection, and diffusion. J. Fluid Mech. 714, 252–279 (2014)ADSCrossRefMathSciNetGoogle Scholar
  36. 36.
    Gray, J.M.N.T., Thornton, A.R.: A theory for particle size segregation in shallow granular free-surface flows. Proc. R. Soc. A 461, 1447–1473 (2005) Google Scholar
  37. 37.
    Phillips, R.J., Armstrong, R.C., Brown, R.A., Graham, A.L., Abbott, J.R.: A constitutive equation for concentrated suspensions that accounts for shear-induced particle migration. Phys. Fluids A 4, 30–40 (1992)ADSCrossRefzbMATHGoogle Scholar
  38. 38.
    Yaroshchuk, A., Zholkovskiy, E., Pogodin, S., Baulin, V.: Coupled concentration polarization and electroosmotic circulation near micro/nanointerfaces: Taylor–Aris model of hydrodynamic dispersion and limits of its applicability. Langmuir 27, 11710–11721 (2011)CrossRefGoogle Scholar
  39. 39.
    Ghosal, S., Chen, Z.: Electromigration dispersion in a capillary in the presence of electro-osmotic flow. J. Fluid Mech. 697, 436–454 (2012)Google Scholar
  40. 40.
    Stone, H.A., Brenner, H.: Dispersion in flows with streamwise variations of mean velocity: Radial flow. Ind. Eng. Chem. Res. 38, 851–854 (1999)CrossRefGoogle Scholar
  41. 41.
    Koch, D.L., Brady, J.F.: A non-local description of advection-diffusion with application to dispersion in porous media. J. Fluid Mech. 180, 387–403 (1987)Google Scholar
  42. 42.
    Pagitsas, M., Nadim, A., Brenner, H.: Multiple time scale analysis of macrotransport processes. Physica A 135, 533–550 (1986)ADSCrossRefGoogle Scholar
  43. 43.
    Mei, C.C., Auriault, J.-L., Ng, C.-O.: Some applications of the homogenization theory. Adv. Appl. Mech. 32, 277–348 (1996)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  1. 1.Department of Mechanical and Aerospace EngineeringPrinceton UniversityPrincetonUSA
  2. 2.Theoretical Division and Center for Nonlinear StudiesLos Alamos National LaboratoryLos AlamosUSA

Personalised recommendations